Do cockroaches eat their shells after moulting? I recently witnessed a cockroach eat its shell after it moulted. Do cockroaches always do this?

Yes, it's fairly common for insects to eat their egg casings, shed skin etc. where possible. In a similar vein, as Alex Stone mentions, spiders often eat their webs too. There's generally plenty of valuable protein and nutrients tied up in there, so why not recycle?